The prime minister is to be chosen by 15 electors. Each is asked to order their preference – first, second and third choice – from three candidates: Nick Clegg, David Cameron and Ed Miliband.

The results are as follows.

Six vote for Cameron first, Clegg second, Miliband third.

Five vote for Miliband first, Clegg second, Cameron third.

And four vote for Clegg first, Miliband second and Cameron third.

Who wins?

Under the UK’s current system, first-past-the-post, Cameron wins – he was the first choice of more voters (six) than anyone else.

But is this fair? Nine of the fifteen voters not only didn’t place him first, they placed him third.

So who else do we choose? Nine of the voters preferred Clegg to Miliband, and nine of the voters preferred Clegg to Cameron. So Clegg wins.

But is this fair? Looking at the first choices, Cameron won six votes and Miliband won five. Clegg only won four, so let’s eliminate him and have a run-off between Cameron and Miliband. In the run-off, six voters would choose Cameron (because six of them placed him ahead of Miliband) but nine would choose Miliband (because they had placed Brown ahead of Cameron). So Miliband wins.

Who wins the election depends on the system you choose to run it. You can make anyone be the winner with the ‘right’ system.

In 1950 Kenneth Arrow published a paper titled A Difficulty in the Concept of Social Welfare. The central result has come to be known as Arrow’s Impossibility theorem. In simple terms it states that in an election with three or more candidates, where voters are asked to rank the candidates, it is impossible to find an overall ranking of the candidates that reflects the preferences of the electorate as a whole. In other words, all electoral systems (of this type) are unfair.